Prove by induction that every positive integer $\displaystyle n$ can be defined as a sum of distinct powers of 2, i.e. as a sum of the subset of the integers $\displaystyle 2^0, 2^1, 2^2$, and so on.

(For the inductive step, consider the case where $\displaystyle (k + 1)$ is even and the case where it is odd.)

I found a variant online on how to prove this via contradiction, but it doesn't use the odd/even cases. I'd prefer it if I had the odd/even induction case version in case it is a requirement.